Sum of Arithmetic Progression/Examples/a0 = i, d = 2+2i

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Example of Sum of Arithmetic Progression

Let $A_n$ be the arithmetic progression of $n$ terms defined as:

\(\displaystyle A_n\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {a_0 + \paren {2 + 2 i} k}\)
\(\displaystyle \) \(=\) \(\displaystyle i + \paren {2 + 3 i} + \paren {4 + 5 i} + \paren {6 + 7 i} + \dotsb + \paren {2 n - 2 + \paren {2 n - 1} i}\)

Then:

$A_n = n \paren {n - 1} + n^2 i$


Proof

\(\displaystyle A_n\) \(=\) \(\displaystyle n \paren {i + \frac {n - 1} 2 \paren {2 + 2 i} }\) Sum of Arithmetic Progression: $a_0 = i$, $d = 2 + 2 i$
\(\displaystyle \) \(=\) \(\displaystyle n \paren {i + \paren {n - 1} \paren {1 + i} }\)
\(\displaystyle \) \(=\) \(\displaystyle n \paren {i + n - 1 + n i - i}\)
\(\displaystyle \) \(=\) \(\displaystyle n \paren {n - 1 + n i}\)
\(\displaystyle \) \(=\) \(\displaystyle n \paren {n - 1} + n^2 i\)

$\blacksquare$


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